How to measure the “well-roundedness” of SE contributors?

Question

Stack Exchange, as we all know it, is a collection of Q&A sites with diversified topics. Assuming that each site is independent from each other, given the stats a user has, how to compute his "well-roundedness" as compared to the next guy? What is the statistical tool I should employ?

To be honest, I don't quite know how to mathematically define the "well-roundness", but it must have the following characteristics:

1. All things being equal, the more rep a user has, the more well-rounded he is
2. All things being equal, the more sites a user participates in, the more well-rounded he is.
3. Whether answer or question doesn't affect the well-roundness

Answer 1

You need to account for similarity between the sites, as well. Someone who participates on StackOverflow and Seasoned Advice is more well-rounded than someone who participates on SO and CrossValidated, who is in turn (I would argue) more well-rounded than someone who participates in SO and Programmers. There are undoubtedly many ways to do that, but you could check overlapping registration to just get a feel for it.

Answer 2

EXAMPLE: say there are three sites, and we want to compare the well-roundedness of the Users A, B, C. We write the reputations of the users across the three sites in vector form:

User A: [23, 23, 0]

User B: [15, 15, 0]

User C: [10, 10, 10]

We would consider A more well-rounded than B (their reputations are both spread out evenly across two sites, but A has more total reputation). Also, we would consider C more well-rounded than B (they have the same total reputation, but C has an even spread across more sites.) It is undecided whether A should be considered more well-rounded than C, or vice-versa.

Let $x_A$, $x_B$, $x_C$ be the above reputation vectors respectively.

We want to measure the "well-roundedness" of a user by a function of their reputation vector $f(x)$. By the above, we would want our function $f$ to satisfy $f(x_A) > f(x_B)$, and $f(x_C) > f(x_B)$.

Any $f(x)$ that is concave and increasing will do the trick.

Two common examples of convex functions are the 'fractional norm'

$$ f([x_1,...,x_m]) = \sum_i x_i^p $$

for $0 < p < 1$.

Taking $p = 1/2$, we calculate

$$f(x_A) = 2\sqrt{23} \approx 9.6$$ $$f(x_B) = 2\sqrt{15} \approx 7.7$$ $$f(x_C) = 3\sqrt{10} \approx 9.5$$

According to the $1/2$-norm, User A would be considered the most well-rounded of the three, by a narrow margin over User C.

Another choice for $f$ is the (scaled) Shannon entropy

$$ f([x_1,...,x_m]) = -\sum_i x_i \log(x_i/c). $$

where $c = \sum_i x_i$.

If we take $f$ to be the scaled Shannon entropy, then we calculate

$$f(x_A) = 46 \log(2) \approx 31.9$$ $$f(x_B) = 30 \log(2) \approx 20.8$$ $$f(x_C) = 30 \log(3) \approx 33.0$$

Measured according the scaled Shannon entropy, then, we would say C is the most well-rounded of the three, and A the second most well-rounded.

EDIT: I originally said the function $f(x)$ had to be convex; the opposite is true.

EDIT2: Added an example in light of whuber's comment.

Answer 3

If you define 'well-roundedness' as 'contributing to many different Stack Exchange Sites,' I would compute some metric of contribution per site. You could use total posts, or average posts per day, or perhaps reputation. Then look at the distribution of this metric across all sites, and compute its skewness in some way that makes sense.

In other words, a 'well-rounded' person would be one who contributes to many different sites, while a 'not well-rounded' person would be one who primarily contributes to one site. You could further improve this by scaling your metric with a user's total across all sites. i.e. someone who's contributed a lot to many different sites should be considered more well-rounded than someone who's contributed nothing to any of the sites. A person who's never used SE isn't very well rounded!

Answer 4

This is a really, really interesting question (indeed I'm somewhat in love with the idea of modelling the stack exchange sites in general).

On the issue of well-roundedness, one way to assess this would be through the tags that particular users tend to answer, and their distribution across sites. Examples may make this clearer.

I am a member on TeX, StackOverflow, CrossValidated and AskUbuntu. Now, I really only contribute to here and StackOverflow, and only about R on Stackoverflow. So,to define well roundedness I would look at a) the amount of tags which two sites have in common (to define similarity across sites) and the extent to which a user answers questions on sites which have little or no tags in common.

If, for instance, someone contributes to Python tags on StackOverflow and cooking, that person is more well-rounded than someone who is answering questions statistical software questions (for instance) on Overflow and stats questions here.

I hope this is somewhat helpful.